Experiments in a cylindrical magnetic shock tube

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EXPERIMENTS IN A CYLINDRICAL MAGNETIC SHOCK TUBE

Thesis by

George C. Vlases

In Partial Fulfillment of the Requirements For the Degree of

Doctor of Philosophy

California Institute of Technology Pasadena, California

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ACKNOWLEDGMENTS

The author wishes to express hie sincere appreciation to Professor H. W. Liepmann. who suggeeted this problem originally and under whose direction the research was carried out. He also wishes to thank Professors J. D. Cole and G. B. Whitham for their fruitful discussions on the interpretation of the experimental results. The help of Mrs. Geraldine Krentler in typing the manuscript and Mrs. Nell Kindig in preparing the figures is also gratefully

acknowledged.

The author is further indebted to the National Science

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iii

ABSTRJ.)CT

An investigation has been conducted with the two-fold purpose of producing very high Mach number shock waves and studying their interaction with an external magnetic field parallel to the shock front. By means of the technique of electromagnetic driving, stable. repro-ducible. outward-going cylindrical shock waves in the Mach number

range from ZO to 100 have been produced and studied.

Theory predicts fundamental differences between the inter-action of a magnetic field with a shock moving into a highly conducting fluid and the interaction of a field with a strong gas -ionizing shock. In the former case a true mhd shock is produced. In the latter the field interacts directly only with the piston and the shock ren1ains an orclinary one. The effect of a conducting wall surrounding the chamber also differs substantially in the two cases.

Detailed experiments have been carried out on gas -ionizing shocks. While the overall motion is' very nearly that predicted by the theory, anomalies have arisen in the details of the flow and are

explained in a qualitative manner.

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TABLE OF CONTENTS

Acknowledgments Abstract

Table of Contents List of :Figures

I. Introduction

A. Scope of the Investigation

B. Production of Strong Shocks, Background Sketch C. The Cylindrical Magnetic Shock Tube

II. Theoretical Predictions of Shock Tube Performance A. Introductory Remarks

B. Shock 1\·:toving into a Conducting Fluid

C. Propagation of Shock Vv aves into a Non-Conducting Fluid

III. Experimental Apparatus and Tecb.niques A. Basic Discharge Chamber

B. Instrumentation

IV. Results and Interpretations for Experiments without Preionization

A. General Remarks and Illustration of the Raw Data B. Detailed Results of ExperirA"lents in Helium and Argon

at 100 1Jo and SOO IJo Initial Pressure 1. Preliminary Remarks

2. Relative Front Positions 3. Kerr-Cell Photographs

PAGE ii

iii iv vi

1

1 2 4

6

6 1 11

17 17 18

20

20

22 22

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v

TABLE OF CONTENTS (Contd.)

4. Wave Speeds 5. P ressure Pulse

6. Magnetic Field Profiles C. Conclusions

V. Preliminary Experiments with Preionization A. General Discussion

B. Preliminary Results

1. ''Second-Half-Cycle" Experiments

PAGE 2.1 2.8 2.8 32.

33

33 34 34 Z. Preionization by Means of a Separate Condenser 34 C. P reionization by Ohmic Heating

D. Conclusion

Appendices

A. Parametric Study of the Cylindrical Magnetic Shock.

B

.

c.

Tube

Sample Calculations of Relevant Plasma Parameters Development of P ressure Probes

D. Design, Construction, and Calibration of Axial Field Coils

E

.

F

.

Circuitry

Basic Equations of Plasma Physics References

Figures

37

37

39 44 48

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LIST OF FIGURES

1. Various Magnetic Shock Tube Co11figurations

2. Schematic View of Cylindrical Magnetic Shock Tube 3, Shock Speed, Greifinger and Cole Solution,

"t =

5/3 4. Piston Speed, Greifinger and Cole Solu~ion.

i

=

5/3

5. Ratio of Shock Speed to Piston Speed, Greifinger and Cole Solution, ?f

=

5/3

6. Oecillograme from Helium at 500 1Jo 1. Osclllograms from Argon at 500 1Jo 8. Oscillogram a from Argon at 100 1Jo 9. Relative Front Positions

10. Relative Front Positions 11. Relative Front Positions

12. Typical Kerr-Cell Photographs in Argon 13. Kerr-Cell Photographs in Argon at 500 IJo 14. Kerr-Cell Photographs in Argon at 100 IJo

15. Kerr-Cell Photographs in Argon at 500 IJo Initial .!?ressure

16.

Effect of Field on Wave Speeds

17, Effect of Field on Wave Speeds 18. Effect of Field on Wave Spe'eds

19. Decay of Pressure Pulse Amplitudes 20. Profiles of rB

6 21. Profiles of ra

9 22. Profiles of rB

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vii

LIST OF FIGURES (Contd.)

24. Profiles of rB 0

25. Profiles of rB 6

26.

Conservation of Flux, A1·gon, 500 1.1. 27. Cross Section of Pressure Probe

28. Discharge Circuits and Typical Timing Sequence 29. Block Diagram of Timing and Triggering Circuits 30. Ignition Firing Circuit

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I. Introduction

A. Scope of the Investigation

Magnetohydrodynamics (mhd) has attracted widespread attention

in recent years. Scientists have become interested in processes

occurring when a conducting fluid interacts with electromagnetic fields.

The study of such processes has been, and will be to an even greater

extent in the future, instrumental in the understanding of many problems

of astro- and atmospheric physics. Engineers have been attracted by

such possible applications as controlled fusion, power generation,

propulsion, reduction of aerodynamic heating, and control of lift and

drag.

The interest in the present experiments lies in the field of wave

motion, and in particular of the motion of strong shocks, where there

are some exciting prospects for the production of very high enthalpy

gases and interesting aspects of the interaction of shock waves with a

magnetic field. These experiments may be envisioned as an mhd analogy

to a classical piston problem in ordinary gasdynamics. The early

experiments (l) centered on the production o£ strong shock waves and

it was found possible to produce ordinary gasdynamics shocks in the

Mach number range from twenty to one hundred which are reproducible, stable. and travel with nearly constant velocity. This Mach number can easily be increased by up to an order of magnitude, as will be discussed below.

Subsequent experiments have been made in which the interaction

of theee strong shocks with a magnetic field parallel to the shock front

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B. Production of Strong Shocks, Background Sketch

During the last decade several methods have been developed

to produce strong shock waves. Conventional shock tubes are capable of producing only relatively low speed shocks, primarily because an

infinite ratio of driver to driven section pressure produces only a

finite (and rather low) "piston pressure", i.e. pressure immediately

behind the shock, for the range of driver to driven section sound speed

ratios available by ordinary means.

Stronger shocks can be produced if either the piston pressure

can be increaaed directly or the driver gas can be substantially heated. The latter technique is the one used in chemically and electrically

driven shock tubes as well as conventional shock tubes. In an electric

shock tube a high current arc is created in the driver section which

heats the gas very rapidly and generates a shock wave. Some of the

earliest work along these lines was done by Fowler(2, 3) and his group at the University of Oklahoma. In recent years the electric shock tube baa been used by many investigators for various purposes such as spectroscopic studies in high energy gases, interaction with external fields, and propulsion.

Alternatively, the "piston pressure" can be increased directly by arranging the Lorentz force resulting from the interaction between the discharge current and a magnetic field in such a way that it aids in propelling the gas; this technique is known as magnetic driving and cannot be accomplished. it). a conventional shock tube.

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electrical conductivity, the heating effect, which is inversely

proportional to the conductivity, decreases. Conversely, magnetic driving becomes more effective as the. temperature increases because the penetration of the field into the fluid decreases; one can think of the piston as becoming less porous. Thus the production of very high speed shocks is most easily accomplished by magnetic driving.

One of the earliest magnetically driven ehock tubes was a

T-tube modified by the addition of a backstrap (4) (Figure 1). In

this device the interaction of the discharge current with the field of the returning current in the backetrap produces a force directed down the tube which helps to propel the gas. Although much of the driving still comes from arc heating, many experimenters have demonstrated that increased shock velocity results from the addition of the backstrap.

Another popular configuration is the conical shock tube as

suggested by Josephson (S). Here all the field, in principle, is trapped

between the current in the gas and the electrodes and acts to force the gas down the tube.

As a final example one might mention the coaxial linear shock tube which has received a great deal of attention, particularly at

A VCO (6). The magnetic field in the annulus acts on the current sheet to product a shock, as illustrated in Figure 1.

'\'ihile each of these devices does produce strong shock waves

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4

co-axial geometry is capable of providing a more continuous push on

the gas but suffers from difficulties in obtaining a uniform current

sheet around the annulus. The current sheet (plasma-vacuum interface)

does not seem to be particularly stable. This fact, along with the

non-uniform magnetic pressure which varies inversely with the square

of the radius across the annulus, makes it difficult to create reproducible,

plane shocks. If the annulus spacing is decreased to make driving field

more uniform the wall losses become correspondingly greater. The

T-tube and conical tube likewise suffer from pi)Or reproducibility and

severe wall effects. Experiments with T-tubes, linear coaxial

accelerators, and transmission line accelerators have previously been

carried out at GALCIT (? • 8).

C. The Cylindrical Magnetic: Shock Tube

The experimental geometry used in the present experiments was

first discussed by Anderson (9), et al. , from a different viewpoint than

the

p~esent

one, a.nd was suggested independently by Liepmann ( lO).

Cul'rent passes thl'ough a solid copper center conductor (Figure 2.) and

returns in a hollow cylindrical arc through the gas. The interaction of

the B

8 field produced by the current in the center conductor with the

returning current produces a Lorentz force, or, equivalently, a magnetic

pressure, which acta radially outward in the manner of a solid cylindrical

piston expanding with time and drives a shock wave into the ambient gas.

The device is thus termed a cylindrical magnetic shock tube. An axial

field Bz can be established prior to the discharge.

0

This device possesses several inherent features which make it

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reproducibility, which makes the experiments feasible. This feature has been extensively studied by Colgate and Furth (ll) and others and will not be reconsidered here. The magnetic piston forms quickly, remains stable, and produces a continuous push on the gas. A great deal of theoretical work h.aa been done on the flow and provides a guide for interpretation of the experimental results. Theory predicts, and experiment verifies, that the shock velocity is constant for a linearly increasing driving current. •l< The outward going front is very accessible for optical and probe measurements. Wall losses (to the top and bottom electrodes) are minimized. The fact that the field is confined entirely within the current sheet means not only that the magnetic driving is effective but also that pickup is greatly reduced.

There are some disadvantages to this geometry which should be pointed out. Since the area of the piston is continually expanding, a continuously increasing current must be supplied to prevent decay. This is in contraet to a linear geometry (co-axial or rail) where in principle a constant current produces a constant speed shock. In addition, contamination from the electrodes is a problem as in any discharge where electrodes are present within the discharge chamber. As a direct result of the cylindrical geometry, the ideal separation distance between the shock and contact surface is half of that in a linear geometry.

In spite of these disadvantages it is felt that the cylindrical shock tube, because of ita stability and efficient driving. is the most successful magnetic shock tube configuration existing at the present time.

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6

II. Theoretical Predictions of Shock Tube Performance

A. Introductory Remarks

In this chapter we shall discuss various theoretical m odels of

the flow to get an idea of expected performance. -The problem of a magnetically driven shock propagating into a region of no field (i.e. the problem studied in the early experiments) is simply a sub-case of the general interaction problem and needs not be discussed separately.

By varying the axial field strength and other parameters the ratio of the ion cyclotron frequency w to the collision frequency "c

ci

can be varied over a wide range, making possible an investigation of

both the "collieionless shock" limit and the collision dominated case

as well as the transition from one to the other. This paper deals

mainly with flows in which w

I

u << 1.

c

1 c

It is assumed throughout that the conductivity in the arc is

sufficiently high that the current runs in a relatively thin annular region,

and the concept of a localized "magnetic piston" therefore has validity. This assumption is fairly well justified a posteriori by the experimental results. Single-fluid theory (mhd) is used except when it is clearly inadequate; recourse is then taken to a more elaborate description.

The nature of the flow will be seen to depend critically on whether or not the fluid ahead of the shock is appreciably conducting.

For that reason the theoretical discussion is divided into two sections,

the first describing the motion when the fluid is initially highly conducting throughout the chamber, and tb.e second describing flows in which a very strong shock propagates into an essentially non-conducting gas.

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attempt at preionization. There are two reasons, however, for

discuss-ing both theoretical rnodels at this time. First, a few c:<.periments have

been carried out in which the gas is rendered highly conducting before

the main discharge, and preliminary results indicate that the 11everywhere

highly conducting" theory is applicable. Secondly, in the absence of

attempted preionizatlon, the gas ahead of the shock may become more or

less conducting due to radiation and electron diffusion through the shock

f ront (l.Z, 13, 14) .

B. Shock Moving into a Conduct;ing Fluid

The simplest theoretical model is a gross balance of rate-of-change of momentum against pressure, suggested by Rosenbluth, and

referred to frequently as snowplow theory. It is assumed that the piston

moves into the fluid and sweeps it up into a very thin, dense layer on the

piston surface. For a cylindrical piston the equation of motion is ( lS)

(1)

where rc is the piston radius, p

0 is the ambient gas density, and

p and p are the magnetic pressures, B 212 1-1o , of the azimuthal

me

mz

and axial fields. In this equation, the radius of the inner conductor and

the ambient gas pressure a1•e assumeci. negligible compared: to the shock

l·adius and magnetic pressure. For the case Bz

=

-o

and I= I 0wt 0

==

(V

I

L)t, where V and L are the initial condenser voltage and

0 0

average* inductance of a slightly damped RLC circuit, equation (1)

• The inductance varies with time as the piston expands. However,

conditions are such that this variation is not important and an average

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8

possesses the conical similarity solution r

=

u t. where

c 0

u

=

0

{

v

2

0~

(2)

and will be called the characteristic snowplow velocity. ForB ~ 0 zo it is found that the solution is again conical. r

=

u t where u

c c c is

related to u and to the Alfven speed in the ambient fluid b by

0 0

l/2

(3)

Clearly uc/ u

0 ~ 1 and the piston is retarded by the axial field.

As is shown in reference 15, snowplow theory is equivalent to

the Newtonian limit in ordinary gasd.ynamics and thua can be expected

to represent the flow only in the case of very strong shocks propagating

into undisturbed fluid (and not for shock reflection. etc.). A shock wave

is termed strong when it propagates with a speed large compared to the

speed of corresponding infinitesimal disturbances. In the present

2 l 1/2

instance. the latter speed is the combined wave speed c

=

(a

0

+

b 0 ) : b0 for a /b <<1. Thus. increasing the axial field while holding the other

0 0

parameters of the problem fixed is equivalent to raising the generalized

acoustic speed and hence to weakening the shock. We therefore expect

a breakdown in the snowplow theory for sufficiently high values of Bz

0 •

An exact solution for the cylindrical magnetic shock tube

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their solution, the current was assumed to rise linearly and the conductivity was taken to be everywhere infinite. The main results of that solution, which of course possesses conical similarity, are

'-summarized below:

(i} Equation 3 predicts the speed of the piston with remarkable accuracy for even very weak shocks. but ia very inadequate for predicting the velocity of any shocks other than very strong. ones.

(U) For a fixed u the shock speed increases and the piston

0

speed ciecreases (according to equation 3) as B increases, zo

accompanied by a decrease in compres a ion of the plasma across the shock. (The rate of increase of shock speed with B is less than the rate of increase of acoustic speed with

30

B so that the shock Alfven Mach number approaches 1. ) zo

(iii) The axial magnetic field increases in passing through the shock in the amount of the density increase, a consequence of flux conservation through a contour attached to the fluid. In the original article by Greifinger and Cole there appeared plots

of shock and piston speeds, and velocity, pressure, density, and magnetic

pressure profiles as functions of the appropriate parameters for the case

¥

=

7/5. The present author is indebted to Dr. Greifinger for extending these calculations to the case ~

=

5/3 for values of parameters pertinent to these experiments. As they are not available elsewhere, the results of the calculations are plotted (in somewhat different form from the original article) in Figures 3, 4, and 5.

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10

which produce the axial field. During the short time of the experiment,

these coils appear to the advancing wave motion somewhat as an

infinitely conducting cylinder which might be expected to influence the

.electric and magnetic fields and hence the fluid motion. It is important

to note that in the case considered here of a very highly conducting fluid

there can be no disturbance (either electromagnetic or fluid-mechanical)

ahead of the shock ( 16); hence the wall has no effect until the first shock

reaches it. An effect does arise, however, from the top and bottom

electrodes. Being highly conducting, they tend to anchor the B field

z

lines and cause them to become convex near the electrodes as the shock

propagates outward. This effect has been minimized by extensive

radial slotting of the electrodes (which is also necessary to establish

the el,ternal field initially (ace Appendix D)).

v.

e

now consider the situation where the conductivity is appreciable

throughout the cham~::>er but not infinite. In this case two new effects

appear, diffusion of the magnetic field lines through the fluid and Ohmic

heating.

At the shock front the diffusion of the induced azimuthal current

is counterbalanced by the non-linear steepening effect familiar in shock

motion, resulting in a steady profile whose thickness is determined by

the transport coefficients. In experiments like these. the width is almost

-1

always of order (~-~ocr ) since the magnetic diffusivity is the largest of

those involved.

The structure of mhd shocks has been studied by Marshall ( 17)

and others. When the shock thickness is governed by conductivity. two

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undergo transition in a length of order (J,.t.cr)-1; for sufficiently low fields there exists a sharp thermodynamic shock at the end of the magnetic field transition region. Thus one sees a sharp Ranl<ine-Hugoniot shock

with a tail on the low pressure side. In a cylindrical geometry, the

jump conditions muat be appropriately modified if the thickness of the

shock is not small compared to the radial position of the shock.

Diffusion of the field into the plasma also occurs at the

plasma-vacuum interface (piston) and there the diffusion depth grows continuously

l/7.. . h • ti h ni

at t s1nce t ere 1s no counterac ng mec a. sm. However, from

considerations of conservation of mass it is clear that the separation between the shock and this piston grows as t so the shock should separate

cleanly from the diffusing piston for sufficiently large t.

The effect of Ohmic heating is to produce a non-constant-velocity

wave system, that ia, to destroy the conical similarity. For example,

if one imagines that moat of the heating occurs near the center during

the initial stages of the discharge whe11 the conductivity is likely to be

quite low, then conditions are approximated for the classical blast wave

solution where the velocity decays as 1/r. Therefore, the

experimentally-determined deviation from constant velocity may provide a measure of the relative importance between heating and magnetic driving.

C. Propagation Gf Shock Waves into a Non-Conducting Fluid

In this section we choose as a model a shock propagating into a

non-conducting fluid in which there is an axial field 13 . The shock ie

. zo

assumed to be sufficiently strong that the gas becomes highly conducting

a short distance behind it. The essential point is that now the field must

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lZ

ordinary Rankine-Hugoniot relationa(lS). This does not mean that there is no effect of the field on the flow, for body forces arise in the region of the piston tending to retard the piston and thus produce a slower

rather than the faster shock produced by the field when the fluid is every-where highly conducting. However, the shocks are strong shocks since their strength depends on the ordinary Mach number. Thus, a snowplow theory remains valid for predicting the m otion.

F'or the case of no wall, equation 3 is again valid for describing the piston path and the flow remains conical. .

In contrast to the model discussed in Section B above, the effect of a conducting wall around the chamber ie now important since electro-magnetic disturbances~ propagate without losses ahead of the shock through· the non -conducting gas.

Consider the idealized problem of a highly conducting, thin, hollow cylindrical piston advancing with velocity u(t) into a vacuum region surrounded by a fixed, highly coilducting cylindrical wall (e. g. a coil producing the field B ). vVe suppose that at t

=

0 the

zo

piston has a radius r

=

r (0) and that a uniform axial field B exists for

e

r < R where R is the radius of the outer cylinder (see sketch), and we investigate what happens to this field as the piston advances.

I I

I

<t

R

v.Lt)

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Assume

..

..

B

=

(0, 0, B(t))

...

E

6 (r, t), 0) E c: (0,

....

..

and that all 9 derivatives vanish, from which div B

=

div E

=

0.

The current flows azimuthally in sheets on the platen (1 ) and

c

on the wall (1 ). The remaining Maxwell's equations are

w

aB

.

< )

ar

=

J..I.Je r, t .

r

8r

=

-From equation 6 there results

Equation 1 yields

B(t)

=

-J..I.l (t); for r < r < R

"... c

E

a

=

-z

!'

.u

'R ' ( t)

r < r

c

E

6

=

-

.!. Z B'(t)

+

f(r t) , r c < r< R

.&ill

r • r>R

where the prime denotes differentiation with respect to time.

(4

)

(5)

(

6

)

(7)

(8)

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14

At the wall and the piston there can exist no electric fields since

the conductivity is assumed to be infinite. This impiiea

E

+

ti

x

B

=

0.

At the piston we have

for r < r (10)

p

or, (r 2B)'

=

0, r

c c

2

B

=

const.

That is., the flux through the area within the piston remains constant 2

in time; the field decreases as 1/r . At the piston we also have the

c

boundary condition

=

0

and at the wall, E

e

=

0 implies

and

g

=

o.

Equations 11 and 13 can be combined to give

or

2 2

[B (R - r ) ] '

=

0 c

B (R2 - rc 2)

=

const. r < r<R

p

(11)

( 12)

(13)

(22)

Thus the flux between the piston and the wall is likewise conserved, that is, the field is compressed between the advancing piston and the wall.

To apply these results to the present experiment one must

reconcile two questions. First. the coils do not close on themselves but are connec~ed to an external circuit. However, the driving capacitors are short circuited (crowbarred) during the passage of the shock across

the chamber.

Secondly, there are gaps between adjacent coila so that some of

the field "leaks out" instead of being compressed between the piston and the wall; the extent of this leakage will be discussed in Chapter IV.

The piston forms initially on the surface of the inner te.flon

insulator so that the field inside this radius is negligible and we may consider r (0)

=

0 for this purpose. Then equation 14 yields

c

and the equation of motion ( 1) takes the form

[ ] [

2 d 2 dx _ x f.!: I (t)

dt X

dt

-

R2p

411'Z

R

z

xz.

0 0

2

z

f.l:(l-x)

where x = r

I R

. For I .- t and B

=

0 the solution is again r

=

u t.

c zo c 0

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(1

6

)

I''or 3

t.

0, however. the piston slows down as it approaches the wall,

zo

even for a linearly rising current. That is, conical similarity is

destroyed in this case by the presence of the conducting wall. The

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16

cone (compare Ref. 1).

In summary, we see that the flow pattern is quite different when the shock advances into a non-conducting fluid rather than into one vvhich

is a good conductor. In the former case the shock speecl increases but

its strength drops; in the latter, the shock travels more slowly but

remains relatively strong.

In the foregoing discassion of expected shock tube performance

the existence of an idealized fluid with scalar conductivity has been assumed. The one fluid model appears adequate to roughly describe the motion of the shock and contact surface. The fact that the fluid is

composed of electrons, ions, and neutrals will, however, become important in understanding some of the detailed experimental results. In particular, the exact mechanism by which the current and field

interact to produce a shock depends on three·fluid effects and is not

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In.

E':xperirnental .Apparatus and Techniques

A short discussion is given here of the experimental apparatus.

Details will be found in appendices C, D, and E'.

A. Baste Discharge Chamber (Figure Z)

The discharge chamber consists of a Pyrex cylinder of 3/811

wall thickness closed by 1/21

' thick glass end plates. Its inside

dimensions are approximately: height,

4",

diameter,

6i".

There are two radial p1·obe inlets in the midplane of the chamber, spaced 60° apart.

A teflon tube of 5/161

' out$ide diameter covers the 3/1611 copper center conductor.

Vacuum down to about one micron is provided by a Cenco mechanical pump used in conjunction with a liquid nitrogen cold trap. All sealing is

done with neoprene 0-rings and the leak rate is on the order of 3 microns per hour. Pressures are measured with a Hastings thermocouple

vacuum gauge whose gauge tubes have been calibrated against a McLeod

gauge for Argon and Helium.

Current for the main discharge is supplied by two 15 J..Lf

capacitors in parallel. The mean inductance of the circuit results in a quarter cycle of 4. 5 J..LS. Most of the tests were run at 12. kv initial

voltage, co:J:responding to a peak current of 123 ka.

The axial magnetic field is produced by discharging a 1000 J..L f

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18

.!3. Instrumentation

On each firing the condenser voltage was monitored as a

function of time with a capacitance type voltage divider for use as a

time reference. It should be noted that, due to the changing circuit

inductance, the time at which the capacitor voltage reaches zero lags

the time of maximum current slightly. This must be taken into

account when reducing the dd.ta. A simpler time reference is obtained

by measuring the rate of change of current with a Rogovsky loop; this

is the method used at present.

At the outset of the experimental investigation it seemed

necessary to develop a direct pressure measuring device in order to

clarify the existing speculation in the literature concerning identification

of observed light fronts with shock or current sheets, etc. Barium

titanate pressure probes were developed which have a rise time of

1/5 IJ. s. They accurately indicate the arrival of a pressure pulse

but, due to pulse reflections inside the transducer, do not faithfully

reproduce the complete pressure history.

Time derivatives of magnetiC fields (f3

6 , Bz) are measured

with small search coila (lmm diameter, 10 turns) encased in a Pyrex

envelope inserted radially into the discharge. For sufficiently high

conductivities, a glass shield is known to have a deleterious effect

(20, 21)

on probe response . However, for the moderate conductivities

encountered in the present experiment, the probes give an accurate

£ f. . (21, 22)

measurement o the telds 1n the plasma . A s a check on

. . (22)

possible electrostat1c effects, a grounded center tap probe used

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and yielded results identical to the single coil probes.

A single-frame Kerr-Cell camera manufactured by

Electro-Optical Instruments. Inc .• is used to take pictures through the side

and top of the chamber at given times after the onset of the discharge.

Exposure times are 1/20 and 1/10 ~ s.

A "run" is accomplished by first discharging the B capacitor

z

bank into the field coils and then firing the main bank after a suitable

delay. A "set" of experiments in a given gas at a particular initial

pressure and condenser voltage consists of pressure, B

8, B z' and Kerr-Cell records for six radial probe stations and five or more values of initial axial field 13 Two probes are used on each run.

zo

The results presented below deal primarily with sets of

experiments in Argon and Helium at 100'"" and 500 ~initial pressures

with all other parameters fixed. Although most of the experiments were done without preionization, some preliminary experiments with

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20

IV. Results and Interpretations for Experiments without Preionization

A. General Remarks and Ulustration of the Raw Data

The purpose of this section is to illustrate some of the general

1•esults of the experiments by presenting oscillograms of the raw data. In subsequent sections a. more detailed description of the results of

three particular sets of runs will be discussed.

Figures 6, 7, and 8 were obtained with a Tektronix 555 dual beam oscilloscope. On each trace the sweep runs from left to right

2

through the 1 em mesh at a rate of 1 ~/em. The discharge occurs

at about 1. 4 to 1. 5 1..1. s after sweep initiation and is responsible for

the pickup at that time appearing on the traces and persisting for

l - 2 1..1. s. (The additional small spurious signals appearing at various

times are also due to pickup.) In general,preamplifier sensitivities vary from one trace

to

the next.

Figure 6 shows the results of a run in .Helium at 500 1.1. initial

pressure., Of particular interest here is a comparison between the

o

0 traces in figures 6a and 6b. In the former the axial field is zero

and it is clear that there is some

B

a

field ahead of the shock and

consequently a current there also. That the signal actt.'lally represents a magnetic field and d.oes not arise from electrostatic effects was proved

by using a double coil and differential preamplifier as described above. The lower trace in figure 6b shows that in the presence of a small axial

field the azimuthal field tends not to extend far forward of the shock.

This is a general result which was observed in all the experiments.

(28)

actually occurs . Z microseconds earlier than the traces indicate due

to the traversal tlme of a stress wave across the probe glass front .

plate.

:Figure 7 was obtained from runs in Argon at 500 ~. In figure

7a the pressure pulse arrives after the peak in the magnetic field

derivative in contrast to the situation in Helium as shown in figure

6;

this anomaly is discussed below. The upper trace of figure 7b was obtained with the magnetic probe oriented to pick up B z. The positive deflection occurring until shock arrival (indicated on the lower trace) indicates compression of the axial field. The ensuing sweeping away

of the axial field after the passage of the piston gives rise to the large negative deflection following the compression.

Figure 8 illustrates runs in Argon at 100 ~. lt can be seen that there is some disturbance on the pressure probe trace before the sharp shock arrives. This is an electrical effect resulting from the

difficulty of shielding the probe completely from electrical fields

(which may become quite large due to charge separation, for example).

It was shown to be an electrical effect rather than a true pressure disturbance by constructing probes of opposite polarity and noting that the !tprecursorfl had the same polarity on different probes while the main disturbance arising from the shock reversed in polarity.

These electrical effects become more predominant in lighter

(29)

2.2

Experiments have been run in air, Argon, and Helium at initial pressures of from 50~ to 500 ~. There seems to be a general tendency

for the shock to be more clearly defined at the higher pressures. Below 100 ~ in Helium the shock is often difficult to pick up, alth.ough the

instrumentation difficulties just described are at least in part responsible.

The magnetic field traces are generally good but also show a tendency to

become le~:Js reproducible, with poorly defined peaks, in Helium at low pressures.

In all experiments the time of arrival of a magnetic peak or pressure pulse is repeatable to a high degree, while the shape and

amplitude of the pulses are somewhat less repeatable (although still quite good in most inetar ... ces),

It is perhaps worthwhile to note that at 100 ~ the mean free path (for neutrals) is 1. 7 mm in Helium and . 7 mm in Argon, long enough to give some structure to phenoinena occurring with a mean free· path length scale, while at 500~ it is one fifth of these values. Thus one expects "pulses" and "discontinuities" to be lesa sharply defined at the higher pressures, which is no doubt in part responsible for the observed behavior below 100~ in Helium.

B, Detailed Results of Experiments in Helium and Argon at lOOJ.lo and 500 J.lo Initial Pressure

In this section emphasis is placed on three sets of experiments which illustrate most of the important features of the flow.

B. 1. Preliminary Remarks. Table I presents some of the relevant parameters (see appendix B for sample calculations). The equilibrium temperature behind the shock (I' 2. ) is calculated from

(30)

Ra.nkine-Ht:goniot relations and equilibrium thermodynamics using

t.lte observed ahock speeds for B

=

0 ancl serves as a. rough estimate

zo

of conditions behind the shock. It is possible, of course, that

thermo-dynamic equilibl'ium is not establishe~ sufficiently quickly to make the

calculation strictly applicable. (Extrapolation of the results of

Petschek and Byron (23) indicates that equilibrium would be reached

(in Argon) in a time between 1

I

2 and 1 1..1o s. The electric field will tend

to keep the electrons hotter, enhancing the ionization rate but tending

to prevent the establishment of true thermodynamic equilibrium.) The

~quilibrium degree of ionization behind the shock (cr.

2 ) is based on

eq T

2 and p2 and is defined as the number of electrons per original

eq

neutral particle. The electl'ical conductivity is obtained from

(24) .

Spitzer's result for fully ionized gases. The diffusion depth of

the driving field

Be

into the gas in the time required to traverse the

chamber, R/u , is also listed, from which it is seen that the "piston"

0

is somewhat diffuse and a clean separation of the shock and piston is

not to be expected.

h1 scanning Table I, one should note that a r"1ajor effect of the

aY..iel field is to decrease the shock strength and hence to produce less

conductivity and consequently greater departures from the idealized

mode.

The state of the plasma is cucll that << 1, that is. the

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Z4

13. Z. Relative F ront Positions. Typical r -t diagrams are

shown in figures

9,

10, and 11. The legend needa a little further

clarification: Circles represent the arrival at a given radius of a

peak in the function B

6(r, t) which for sharp peaks (ideal flow)

correspond closely to the maximum current density j , i.e. to the

z

arriv-al of the magnetic piston at the distance r

1. The symbols

labeled 11 B null11 refer to the null in

B (r

1, t) which occurs when

z

z

compression oi the axial field at a given radius is complete. This,

according to the theoretical discussion given above, should correspond

closely to the shock position. Finally, the solid circles represe1'1t the

radius of the luminous front. 'vVhere deviations fro:::n cylindrical

symmetry were observed, the positions of the leit- and right-hand

sides of the front are both shown.

These figures illustrate the relative positions of the various

fronts. In Helium at 500 IJ. (figure 9) the shock precedes the piston as

one expects. This is true both with and without the axial field. The

separation between the fronts changes little with field strength.,

indicating a non-mhd shock flow. For no field the light :front is very

near the shock, while it lags slightly when a field is applied. Figure 10

illustrates a result which at first appears somewhat paradoY.ica.l. For

For B

=

0, the piston appears to be slightly ahead of the shock, while zo

for B

=

4950 the 11piston•• is definitely ahead of the shock. It should

zo

be recalled tha~ the 11

piston11

is really a relatively thick annular region

in which ·volume currents flow, rather than a cur rent sheet at a w

ell-defined location. · Thus, the shock is located ~ear the rear of the Ct.\rre

(32)

(figure 9). Figure 11 shows that a similar tendency for the piston to

precede the shock occurs in Argon at 100 IJo , although it is 110t so pronounced.

A possible explanation of the piston niJrececling" the shock can

be given as follows. The electrons are the primar}r recipients oi the

Lorentz force acting on the gas because they carry most of the current.

Thuo they tend to move away from the ions which are pulled along

behind them by the electrostatic force arising fron.1 the charge separation which occurs. I.e:; the ions are heavier, they are more effective in transferring energy to the neutrals a::1d the principal

11ahockn disturbance is associated with the ions. The separation

which can be maintained hetv1cen electron:J and ions depends h'l part on the relative ease with which the electrons and. ions can move through

the neutrals, as well as on the product of the Debye length times the

Miach number. (Since the electron concentration varies by several orders of magnitude in passing through the shock region, the Debye

length io difficult to determine even approxiraat~ly.). Now the

electron-atom collision crosG scctioro for Heliu::."n and .l~.rgon arc quite different in the energy range of 1 to 3 e. v. which is characteristic of these experiments. The elect:ron·-aton1 cross section for ·Argon as a

function of energy displays a vel·y pronounced dit~ at 1 e. v.

(Ramoauer effect) to a value less than 1/6 that o:£ Helium, whose

electron-atom erose section is approximately independent of energy • hi (·:.o ' ·-· l (ZS)) Rlb 1 tl "

1n t s range ~-·~assey ana tsur 1op . ..~ ave e. v. 1e .. ~rgon

croas section increuse3 rapidly, becon"Jing equal to that of Heliurn at

(33)

26

temperature in Argon at 500 1-1 for no axial field is about Z e. v. The presence of a field slows the ahock (s~e below) and moves the tempera-ture toward 1 e. v., where the cross section is the smallest; this is precisely where the largest separation of the fronts occurs. The experiments in Argon at 100 1.1o correspond to a cross section

inter-mediate between that of the experiments in Helium at 500 IJo, where the piston lags as in a normal shock-piston flow, and Argon at 500 1-1 . As seen in figure 11, the relative positions of the fronts tn this case are consistent with the above reasoning.

The foregoing explanation is certainly a11 over-simplification of a complicated phenomenon, but does suggest the importance of the Ramsauer effect in the piston mechanism. Obviously a great deal of

work remains to be done before the process whereby a magnetic piston drives a gas and produces a shock wave is thoroughly understood.

B. 3. Kerr-Cell Photographs. Several interesting features of the flow are shown in figure 12. As is evident, the light front is straighter and more clearly defined at high initial pressures. In

Argon, with no axial field applied, amall offshoots or flares of

luminosity are present which disappear when even a slight initial field is used. In general, the presence of the axial field is to enhance the stability of the luq1inous front.

A slight tendency waa noted. particularly in Argon in the absence of a B field, for the radius of the cylindrical light front at a given time

z

(34)

corresponds closely to the ionization front and shock, as is apparent from figures

9.

10, and 11.

Figures 13 and 14 show quite well the effect of the axial field on the radius of the luminous front and also give a good overall view of the difference between the high and low initial pressure ex.periments. (The extraneous light at the edge of the photographs ie due to reflection in the Kerr-Cell caused by the omission of a collimating lens from the optical system.) Photographs were aleo taken in Helium, but due to the lower intensity of visible radiation emitted the pictures do not show enough contrast for good reproduction.

Figure 15 was taken through the top of the chamber and illustrates the outstanding cylindrical symmetry of the flow.

In figures lZ, 13, and 14, some curvature of the light front near the electrodes due to the anchodng of the field linea is observed,

however, the effect ia quite small.

B. 4. VJave Speeds. Figures 16, 17, and 18 illustrate the

influence of the axial field strength on the motion of the shocl' and piston. In each case, the shock is slowed by the axial field. This is fairly

conclusive evidence that the shock is an ordinary one propagating into an essentially non-conducting fluid, as discussed in Chapter II. The

increasing deviation from conical similarity as the field increases indicates further that the effect of the conducting wall (coils) is being felt. Note

(35)

28

experimental points is obtained.

B. 5. Pressure Pulse. Ae the shock slows, the pressure jump

must drop. Calibration of the pressure probes in a.n ordinary shock

tube shows that the output is roughly proportional to the reflected shock

pressure jump. For etrong shocks, this is proportional to the square

of the shock Mach number, so that output falls rapidly as the shock

speed drops. Comparison of figure 19 with figures 16 and 17 shows

this to be qualitatively experimentally verified.

B. 6. Magnetic Field Profiles. The macnetic r;:robe traces can

be integrated to give Be (r

1, t), where r1 is the probe location on a

given run. ~.Y erose plotting the results for several runs one obtains

magnetic field profiles B(r, ti) for several times during the propagation of the wave aystem across the chamber.

At any given instant the azimuthal field,

s

8, should decay as 1/ r

in the region between the center conductor and the piston and drop to

zero through the piston. Therefore a plot of rB

6 vs r for a given t

should resemble a negative step; i.e. rB~

=

conet for r < r < r , and

v 0 c

rB

6

=

0 for r> rc for the idealized case of a current sheet piston. 1 &(rBe)

Since J.

z

= -

~r

ar

,

a current flows wherever the slope is non-zero.

Such plots are presented in figures 20 through 25 (the times

denoted on the curves should be increased by 0, 2~ sec.). As above,

examination of the flow details reveals some anomalous results.

The magnetic field profiles shown in figure 20 are very similar

to the step described above. As time increases the position of the

current front advances nearly to the wall. Figure 21 shows that in the

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and the current diffuses slowly to the wall.

\Ve next consider figure 24. The curves indicate that as the

driving current builds up, additional currents begin to flo"" in the

"vacuum" region behind the piston, in the direction of the current

in the center conductor near the center of the chamber, and in the

opposite direction further out.

One's first inclination i5 to distrust the results for two x·easons: first because the experimental errors are the largest near the center,

_ and secondly, because the curves depend on the assumption of detailed

repeatability of the magnetic traces. Repeated experiments, however,

indicate the same qualitative behavior.

There are several comments to be made: First, in order for a current to flow there must be some plasma left behind the piston. The density can be quite small since the conductivity of a fully ionized gas is essentially independent of density (24). Thus it is possible that the

mn.as oi plasma. left behind the piston iG small enough that the total momentum balance is not significantly disturbed. i.e. the snowplow theory given above remains valid. The "leakiness11 of the piston depends

on the conductivity, which is lower for Helium than for Argon for the

col'lditions of the experiments (Table I).

Although j can get w~ry large near the ce11te1' (larger thall in the

z

piston, for example), the current carrying area is relatively small and

the total 11induced additional currentn never exceeds 15 per cent of the main drive current.

(37)

30

interesting to note from Table I that the conductivity here ia also intermediate between those of tlle othe1· experiments.

One notes also that the "induced current" is reduced in the presence of an axial field, an effect which is apparently due to the tendency of the a:d.al field to inhibit plasma motion.

It remains to be shown that an electric field exists which could produce auch a current. Consider the idealized problem of the electric field between the elements oi a co-axial conductor separated by vacuum . The current is assumed to rise linearly with time and the boundary conditions are

E = E (t) at r

=

r 1' z zl

t

f}, E

E

=

E (t) at r

=

,..

z

Z

z

rz·

Y..,

E

=

0 at z

=

z

.

r 0

l"-t

Under such conditions t..~e electric field i:3 given by E {t)

+

E (t)

zl

Zz

.tn

.t

n rz/rl

=

(38)

interval r

1 < r <r 2 and which could therefore give rise to the observed current pattern.

The fields E and E are simply equal to j

I

cr in the two

z

1

z

2

z

conductors. Rough calculation ahows that the values of E' and E

. · zl ~z

computed for the conditione of the experimenta are somewhat less than that required to produce the observed results. Alternatively, the conductivity of the plasma would have to be a factor of five or so greater than the equilibrium conductivity listed in Table I; this is not inconceivable because the electric field tends to maintain the electron temperature at a higher value than that given by equilibrium thermo-dynamics.

It is felt that these arguments give a general indication of the observed phenomena. Ho\...,ever, because of the inaccuracy of the

measurements near the center conductor and the over simplified model which wae c~1oaen, further attempts at quantitative deocription on this level are not justified. Clearly, more work muat be done before a detailed understanding is achieved.

Finally, we retur1• to the point of conservation of flux as

discussed in Chapter

·ux.

In order for flux to be conserved in the chamber as the piston advances, we must have

coil

Jr

(Bz - Bz 0

) dr

=

0 center

at any given time. In figure 26 the quantity r(B - .l3 ) is plotted z z

0

(39)

32

but that some flux is lost from within the chamber as the piston

advances.

C. Conclusions

The experiments performed without preionization indicate that an ordinary shock wave propagates into the ambient fluid at a rate

governed by the speed of the magnetic piston, wb.ose motion is retarded

in the presence of an axial magnetic field.

The gross motion is very well predicted by snowplow theory. In examining the details, however, One encounters anomalies which

have thus far been explained only qualitatively. A thorough understanding

(40)

V. Preliminary Experiments with Preionization

A. General Discussion

From the discussion in the foregoing chapter it is clear that the gas ahead of the shock wave is not sufficiently conducting for an mhd shock to be produced. In order to create a shock whose thi-ckness is small compared to the chamber dimensions the ambient gas must have

4 5

a conductivity of at least 10 to 10 mhos/meter.

Preionization by means of r-f induction has been used in many experiments. In most of these, however, the goal bas been simply

to·produce some electrons so that the main discharge proceeds smoothly. Measurements have shown that conductivities produced by r-f heating in general are less than

that

required here. In addition, the tendency of the electrodes in the present experiment to e>;clude high frequency

external fields makes it very unlikely that either the desired magnitude or uniformity of the conductivity could be attained. so that r-f heating

appears to be unsatisfactory for the present purposes.

One of the most effective ways to heat a gas h by passing a strong shock or repeated strong shocks through it. In the cylindrical shock tube a shock wave is generated each time the condenser rings through a half-cycle. Thus the first shock can be used to heat the gas and measurements made on the second, for example. In this method the state of neither the axial field nor the fluid is known since the second shock comes so soon after the first; conditions are probably very non-uniform. Some meal!lurementa have been made on the ''second-half-cycle shock" and will be discussed below.

(41)

34

constructed which utilizes two capacitor banks firing through the same

electrodes (Fig. 2.8). The second bank can be delayed an arbitrary

amount with respect to the first, and thus two series of shock waves can

be generated. The firat is used to render the gas conducting. After

the gas comes to some sort of equilibrium the second bank is fired and

meac:mrementa are made on the first shock resulting therefrom.

B. Preliminary Results

B. 1. "Second-Half-Cycle" Experiments. Measurements of the

wave originating from the second-half-cycle oscUlation of the main

condenser (separate preionization bank disconnected) in the absence of

an external magnetic field show that the shock forms immediately and

travels outward, with the snowplow velocity based on original ambient

pressure, to a radius of approximately Z". Beyond that distance it

attenuates rapidly and disappears before reaching the wall. Preliminary

rune with an axial field failed to give conclusive evidence of a true mhd

shock. Because of the uncertainties involving the initial conditione for

the second shock, this method of preionization has not been further

pursued,

B. 2.. Preionization by Means of a Separate Condenser. Initial

tests without an axial field were made with a series inductance of 7. 5 1.1. h,

(see Fig. 2.8 and App. E). with a resulting ringing time for 'the

pre-ionization bank of 47 1.1. s. It was found immediately that initiation of the

main discharge at a time other than that of a null in the preionization

current resulted in the main arc starting at the place where the

(42)

94 ~s after the preionization bank, corresponding to two full cycles

of preionization current.

For radii greater than l" the pressure and magnetic traces are

extremely reproducible. Within 2" of the center a pressure pulse in·

general can not be detected and the magnetic traces are not very well

defined or repeatable. Kerr-Cell photographs reveal that the luminous

front initiates near the center and spreads very quickly to a radius of

1" to 1 1/211

, after which a stable front forms and tra vela at the

snowplow speed. However, even for radii greater than 2" the luminous

front remains an approximately constant distance ahead of the shock and

magnetic fronts.

One can explain this behavior qualitatively as follows. The

electrodes are initially nearly equipotential surfaces and apply a

uni!orm'·electric field across the gap. The discharge tends to originate

at the center for reasons of mlnim~m inductance, after which the

electric field becomes small. If the plasma is very highly conducting

the field and current diffuse very little and the behavior is nearly

ideal, i.e. a ''piston'' forms. If the gas is initially non-conducting the arc is again confined to a narrow region where the conductivity is

non-zero and an effective piaton is formed. In the intermediate case,

however, the diffusion speed is initially great and the current diffuses

rapidly into the moderately conducting plasma. and distributed body

forces begin acting to form a shock, which then overtakes the diffusing

piston. Thus the conductivity must be made quite high to get a

piston-like behavior in the initial stages.

(43)

36

removed 'from the preionization circuit, resulting in a ringing period of 16 ~ s. The capacitor is allowed to oscillate through six complete cyclea, after which it is nearly completely damped. This sends many strong shocl~ waves through the chamber, heating the gas to a high degree. The main discharge is then initiated. An upper bound to the energy deposited in the gas is just the energy stored originally in the capacitor (most of the dissipation is in the gas so that this should be a fairly good estimate). This amounts to 85 ev/pa.1•ticle at 500 ~ initial pressure, and 420 ev/particle at 100 ~ , for a 7. 5 ~ f capacitor charged to 12 kv. This energy is more than sufficient to attain the desired

conductivity.

There will, of course, be losses, and these must be calculated or measured. Most of the loss is due to radiation. Calculations show that losses from Bremsstrahlung are negligible but that those from optical radiation may not be. In future experiments the electron temperature will be measured, either spectroscopically or with Langmuir-type probes, to determine the time at which conductivity is a maximum.

One set of runs has been carried out with this circuit, in Argon at 500 ~. in which pressure and magnetic probes were placed at a fixed radius while the axial field was increased in successive runs in steps of 500 gauss. A sharp shock exists at the lowest field strengths. _1\t

higher !ield3 the tendency is for the leading edge of the shock to. deteriorate while remaining sharp at the back. At the highest fields only a slow

pressure transition is observed. This is in qualitative agreement with

(44)

to move slightly ahead of the piston as the field ls increased. Although these initial results are encouraging, a great deal of work remains to

be done before any definite ccmclusiona can be reached.

C. Preionization by Ohmic Heating

One final method of preionization, involving an ordinary

discharge between the top and bottom electrodes without the use of the center conductor·, · should be noted. This method, in which there is now no outward Lorentz force (indeed, the Lorentz force is radially inward, i.e. it pinches, but is rather small.for small currents and no outer return cylinder), has been used in "inverse pinch experiments'• at the

' . (26)

University of llhno1s , and may be effective in the present

experiments for the following reasons. Electrical conductivity depends primarily on electron temperature(Z-4), which in a discharge tends to remain higher than ion temperature due to the poor electron-ion energy transfer cross section. Thus it may be possible to establish a uniform discharge throughout the chamber with a current small enough that the

Lorentz force is negligible, but large enough to ionize the gas to a high

degree and heat the electrons sufficiently to achieve the required

conductivity. This would have the additional advantage of not greatly disturbing the gas density distribution and initial axial field.

D. Conclusion

Heating the gas by means of multiple strong shocks eeeme to

hold promise as an effective way to produce the initial conductivities

(45)

38

encouraging, but inatrun'lentation oust be developed to rneasure

conductivities achieved by the preionization bank. A direct Ohmic

heating discharge without the use of the center conductor may be an

(46)

Appendix l\. Parametric Study of the Cylindrical lv1agnetic Shock Tube

The purpoae o£ this appendix is to illustrate the dependence of various quantities such as the shock speed, Mach number, enthalpy, conductivity, testing time, etc., on the relevant parameters, which include the circuit inductance, capacitance, and voltage and the type of gas and ita initial pressure.

Remarks will be confined to the case of no axial field. We will assume the shocks are strong, in which case snowplow theory is valid.

The characteristic shock speed, which we have seen to be e::q>erimentally verified, is

_

~.~o_o..---}

1

I 4

,...,

2

8w p 1

(k T )1/4

1

1/4

1/4

pl m

where the subscript 1 denotes conditions ahead of the shock** and the ambient gas has been assumed perfect,

Thus, to achieve the highest velocity, one uses a high voltage and low inductance and a light gas at low initial pressure. The initial

temperature, T

1, will be taken to be room temperature.

( . .'\. 1)*

(A. 2)

The capacitance does not enter into equation 1 directly because the assumption of a linearly rising current is equivalent to aesurning a ·

*Equation numbers within an appendix include the letter of the appendix.

(47)

40

constant voltage source (very large capacitance). It has been found

that if the quarter cycle of this discharge is greater than or equal to the time required for a wave to traverse the chamber, i.e.

i

-{'LC

~

~

·

then the front travels with nearly constant speed all

0

the way across the chamber. The capacitance should therefore be chosen large enough that this inequality holds for the smallest

valuee of u which will be encountered·in the e:>:periments, that is

0

Diffusion of the driving field into the plasma is minimized

(A. 3)

by making the conductivity behind the shock as high as possible. The

conductivity depends on the temperature and hence on the enthalpy

and pressure, which are functions of the shock Mach number.

The speed of sound in the ambient gas is

so that the Mach number is

u ..! 1/4

M -:

-

0 ~ v~ - - ! m

1/4

. s - a.l L~

p

Thus, for a given V, L. and p1• the Mach number ie higher for a heavier gas than for a light one.

The enthalpy and pressure behind the shock are related to those in front by the strong shock relations ( 27)

"1

+

1 2 2

h2/h

1 ~ r-.18 (1 - l/r1 )

2

(48)

where 1"J is the density ratio and is of the order of 10 for strong

gas-ionizing shocks. Since h

1

=

(5/2) kT1, we have

mh

2 A

-

MG 2 (

~

)'~i+l

2

kTl

2

..v.Ms (A. 5)

and

pl (S'l Ms 2 -- p l lvi3

2

(A. 6)

p ~ 2

Notice that mh

2 is the enthalpy in region 2 per .original particle

and that Pz is the actual pressure._ A fitudy of the .Mollier chart shows

that in the region where (multiple) ionization is occurring theoe two

quantities must be maximized to maximize the temperature.

Combining (4) with (5) and (6) we find

(A. 7)

(A. 8)

Thus a high initial pressure tends to lower the enthalpy per particle

bttt to increase the pressure behind the ehock. One can not therefore

draw any general conclusions from (7) and (8) as to whether a high or

Figure

40 40 12o =O 20 FIG. 3
40 40 12o =O 20 FIG. 3 p.89
FIG. I 6
FIG. I 6 p.102
FIG. 2~ PROFILES OF
FIG. 2~ PROFILES OF p.109
FIG. 24 PROFILES OF
FIG. 24 PROFILES OF p.110
FIG. 25
FIG. 25 p.111
FIG. 30
FIG. 30 p.116

References

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